Abstract

It is proved that if an uncountable cardinal κ has an ineffable subset of weakly compact cardinals, then κ is a weakly compact cardinal, and if κ has an ineffable subset of Ramsey (Rowbottom, Jónsson, ineffable or subtle) cardinals, then κ is a Ramsey (Rowbottom, J\'onsson, ineffable or subtle) cardinal.

Highlights

  • Large cardinals imply the existence of stationary subsets of smaller large cardinals

  • For instance weakly compact cardinals have a stationary subset of Mahlo cardinals, measurable cardinals imply the set of Ramsey cardinals below the measurable cardinal κ has measure 1 and Ramsey cardinals imply the set of weakly compact cardinals below the Ramsey cardinal κ is a stationary subset of κ

  • R ⊆ κ is an ineffable subset of κ if for every sequence Sα | α ∈ R such that Sα ⊆ α for α ∈ R there exists T ⊆ R a stationary subset of κ such that for every α, β ∈ T, α < β, Sα = α ∩ Sβ

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Summary

Introduction

If κ is an almost ineffable cardinal, the set of weakly compact cardinals below κ is an almost ineffable subset of κ (see [1]). In [1], Theorem 4.1, it is proved that if X is an almost ineffable subset of κ, the set {α ∈ X : α is a Π1n-indescribable cardinal} for every n < ω, is an almost ineffable subset of κ.

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